Overview
This lecture covers the concepts of series, including finite and infinite series, partial sums, and expressing series using Sigma (summation) notation.
Series: Definition and Types
- A series is the sum of the terms in a sequence.
- Adding sequence terms (e.g., 1, 3, 5, 7, 9, 11) forms a series (e.g., 1 + 3 + 5 + 7 + 9 + 11).
- An infinite series is the sum of all terms in an infinite sequence.
- A finite series (partial sum) sums only a specified number of terms from a sequence.
Partial Sums (Finite Series)
- Sβ denotes the sum of the first n terms of a sequence.
- Example: Sβ for sequence 1, 3, 5, 7, 9, 11 is 1 + 3 + 5 + 7 + 9 + 11.
- Calculating partial sums: Add the terms up to the desired position (e.g., Sβ = first two terms).
- Examples:
- Sβ for 5, 15: 5 + 15 = 20.
- Sβ for 5, 15, 25, 35: 5 + 15 + 25 + 35 = 80.
- Sβ for 2, β6, 18, β54: 2 + (β6) + 18 + (β54) = β40.
- Sβ for extending above: sum up first six terms.
Sigma (Summation) Notation
- Sigma notation provides a concise way to represent series.
- Uses the symbol β (Greek letter Sigma) with lower and upper limits to denote start and end indices.
- General format: β (expression) with index from lower to upper limit.
- Example: β (3k) from k=1 to 4 is 3Γ1 + 3Γ2 + 3Γ3 + 3Γ4.
- The variable can be any letter (commonly k, i, or j).
Examples: Expressing Series in Sigma Notation
- 1 + Β½ + β
+ ΒΌ + β
: β (1/k) from k=1 to 5.
- 1Β² + 2Β² + 3Β² + ... + nΒ²: β (kΒ²) from k=1 to n.
- 1 + (β3) + 5 + (β7): β [ (β1)^(k+1) Γ (2kβ1) ] from k=1 to 4.
Key Terms & Definitions
- Series β The sum of the terms of a sequence.
- Infinite Series β A series with an infinite number of terms.
- Finite Series (Partial Sum) β The sum of a fixed number of terms from a sequence.
- Sigma Notation (β) β A notation to represent the sum of terms in a sequence, using an index and limits.
Action Items / Next Steps
- Practice writing given sequences and their sums in Sigma notation.
- Calculate partial sums for various sequences as demonstrated.